Section.1: Exponent Properties Ojective: Simplify expressions using the properties of exponents. Prolems with exponents cn often e simplied using few sic exponent properties. Exponents represent repeted multipliction. We will use this fct to discover the importnt properties. World View Note: The word exponent comes from the Ltin expo mening out of nd ponere mening plce. While there is some dete, it seems tht the Bylonins living in Irq were the rst to do work with exponents (dting ck to the 2rd century BC or erlier). Exmple 1. Simplify. 2 ()() 5 Expnd exponents to multipliction prolem Now we hve 5 0 s eing multiplied together A quicker method to rrive t our nswer would hve een to just dd the exponents: 2 = +2 = 5 This is known s the product rule of exponents. Product Rule of Exponents: m n = m+n The product rule of exponents cn e used to simplify mny prolems. We will dd the exponents on like ses. This is shown in the following exmples. Exmple 2. Simplify. 2 6 Sme se; dd exponents 2 + 6 + 1 9 Exmple. Simplify. 2x y 5 z 5xy 2 z 10x 4 y 7 z 4 Multiply 2 5; dd exponents on x; y nd z Rther thn multiplying, we will now try to divide with exponents. 77
Exmple 4. Simplify. 5 2 Expnd exponents Divide out two of the 0 s Convert to exponents A quicker method to rrive t the solution would hve een to just sutrct the exponents: 5 2 = 5 2 =. This is known s the quotient rule of exponents. Quotient Rule of Exponents: m n = m n The quotient rule of exponents cn similrly e used to simplify exponent prolems y sutrcting exponents on like ses. This is shown in the following exmples. Exmple 5. Simplify. 7 1 7 5 Sme se; sutrct exponents 7 8 Exmple 6. Simplify. 5 5 c 2 2 c Sutrct exponents on ; nd c 5 2 2 2 c A third property we will look t will hve n exponent expression rised to second exponent. This is investigted in the following exmple. Exmple 7. Simplify. ( 2 ) Notice 2 three times 2 2 2 6 Add exponents Our solution A quicker method to rrive t the solution would hve een to just multiply the exponents: ( 2 ) = 2 = 6. This is known s the power of power rule of exponents. Power of Power Rule of Exponents: ( m ) n = mn 78
This property is often comined with two other properties which we will investigte now. Exmple 8. Simplify. () ()()() Notice () three times Three 0 s nd three 0 s cn e written with exponents A quicker method to rrive t the solution would hve een to tke the exponent of three nd put it on ech fctor in prentheses: () =. This is known s the power of product rule of exponents. Power of Product Rule of Exponents: () m = m m It is importnt to e creful to only use the power of product rule with multipliction inside prentheses. This property does NOT work if there is ddition or sutrction. Wrning! ( + ) m =/ m + m These re NOT equl; ewre of this error! Another property tht is very similr to the power of product rule is considered next. Exmple 9. Simplify. Notice the frction three times Multiply frctions cross the top nd ottom nd use exponents A quicker method to rrive t the solution would hve een to put the exponent on every fctor in oth the numertor nd denomintor: known s the power of quotient rule of exponents. m Power of Quotient Rule of Exponents: m = m =. This is 79
The power of power, product nd quotient rules re often used together to simplify expressions. This is shown in the following exmples. Exmple 10. Simplify. (x yz 2 ) 4 Put exponent of 4 on ech fctor; multiply powers x 12 y 4 z 8 Exmple 11. Simplify. ( )! 2 c 8 d 5 Put exponent of 2 on ech fctor; multiply powers 6 2 c 16 d 10 As we multiply exponents, it is importnt to rememer these properties pply to exponents nd not the ses. An expression such s 5 does not men we multiply 5 y ; insted we multiply 5 three times, 5 5 5 = 125. This is shown in the next exmple. Exmple 12. Simplify. (4x 2 y 5 ) Put the exponent of on ech fctor; multiply powers 4 x 6 y 15 Evlute 4 64x 6 y 15 In the previous exmple we did not put the on the 4 nd multiply to get 12. This would hve een incorrect. Never multiply se y the exponent. These properties pertin to exponents only nd not the ses. In this lesson we hve discussed 5 dierent exponent properties. These rules re summrized in the following tle. Rules of Exponents Product Rule of Exponents Quotient Rule of Exponents Power of Power Rule of Exponents Power of Product Rule of Exponents Power of Quotient Rule of Exponents m n = m+n m = n m n ( m ) n = mn () m = m m m m = m 80
These ve properties re often comined in the sme prolem. Often there is it of exiility s to which property is used rst. However, order of opertions still pplies to prolem. For this reson it is the suggestion of the uthor to simplify inside prentheses first; then simplify ny exponents (using power rules); nd nlly simplify ny multipliction or division (using product nd quotient rules). This is illustrted in the next few exmples. Exmple 1. Simplify. (4x y 5x 4 y 2 ) In prentheses simplify using product rule; dd exponents (20x 7 y ) With power rule; put three on ech fctor; multiply exponents 20 x 21 y 9 Evlute 20 8000x 21 y 9 Exmple 14. Simplify. 7 (2 4 ) 7 (8 12 ) 56 15 Prentheses re lredy simplied; use power rule Use product rule; dd exponents; multiply numers Exmple 15. Simplify. 10 4 2 4 2 0 7 4 2 4 2 15 2 Simplify numertor with product rule; dd exponents Now use the quotient rule to sutrct exponents Exmple 16. Simplify. m 8 n 12 (m 2 n ) Use power rule in denomintor m 8 n 12 m 6 n 9 m 2 n Use quotient rule Our solution 81
Exmple 17. Simplify. 2 (2 4 2 ) 6 5 7 2 (8 12 6 ) 6 5 7 24 1 8 6 5 7 2 2 2 Simplify inside prentheses rst; use power rule in numertor Simplify numertor; use product rule Simplify; use quotient rule (4 8 ) 2 Prentheses re simplied; use power rule 16 16 2 Clerly these prolems cn quickly ecome quite involved. Rememer to follow order of opertions s guide; simplify inside prentheses rst; then use power rules followed y the product nd quotient rules. 82
.1 Prctice Simplify ech expression. 1) 44 4 4 4 ) 4 2 2 5) m 4mn 7) 2m 4 n 2 4nm 2 9) ( ) 4 11) (4 4 ) 2 1) (2u v 2 ) 2 15) (2 4 ) 4 17) 45 4 19) 2 21) nm2 n 2) 4x y 4 xy 25) (x y 4 2x 2 y ) 2 27) 2x(x 4 y 4 ) 4 2x 7 y 5 29) x y 4x 2 y (2x) 2 1) x 2y 17 ) (2x 2 y 4 ) 4 5) 2m n 4 2m 4 n 4 mn 4 7) 2xy5 2x 2 y 2xy 4 y 9) q r 2 (2p 2 q 2 r ) 2 2p 41) zy z x 4 y 4 x y z 4 4) 2x2 y 2 z 6 2zx 2 y 2 (x 2 z ) 2 2) 4 4 4 4 2 4) 2 6) x 4x 2 8) x 2 y 4 xy 2 10) (4 ) 4 12) ( 2 ) 14) (xy) 16) (2xy) 4 18) 7 20) 4 22) x2 y 4 4xy 24) xy 4xy 26) (u 2 v 2 2u 4 ) 28) vu5 2v uv 2 2u v 0) 27 2 4 2 4 2) 22 2 7 ( 4 ) 2 4) yx2 (y 4 ) 2 2y 4 6) n (n 4 ) 2 2mn 8) (2y x 2 ) 2 2x 2 y 4 x 2 40) 2x4 y 5 2z 10 x 2 y 7 (xy 2 z 2 ) 4 42) 2q p r 4 2p (qrp ) 2 4 8
1) 4 9 =262,144 2) 4 7 =16,84 ) 2 4 =16 4) 6 =729 5) 12m 2 n 6) 12x 7) 8m 6 n 8) x y 6 9) 12 = 51; 441 10) 4 12 =16,777,216 11) 4 8 =65,56 12) 6 = 729 1) 4u 6 v 4 14) x y 15) 16 16 16) 16x 4 y 4 17) 4 2 = 16 18) 4 = 81 19) 20) = 27 21) m 2 22) xy 4 2) 4x2 y 24) y2 4 25) 4x 10 y 14 26) 8u 18 v 6 27) 2x 17 y 16 28) uv 29) x2 y 6 0) 42.1 Answers 1) 64 2) 2 ) y 512x 24 4) y5 x 2 2 5) 64m 12 n 12 6) n10 2m 7) 2x 2 y 8) 2y 2 9) 2q 7 r 8 p 40) 4x 2 y 4 z 2 41) x 4 y 16 z 4 42) 256q 4 r 8 4) 4y 4 z 84